148 SAMPLED-DATA CONTROL SYSTEMS 
In view of the fact that D(z) is generally implemented by active ele- 
ments, it is theoretically capable of producing any desired physically 
realizable K(z). For this reason, considerable attention must be given to 
the desired prototype forms of K(z) which must be sought. As will be 
seen later, there are some subtle limitations on K(z) which must be 
examined before design procedures can be outlined. 
7.3 ‘‘Minimal’’ Over-all Prototype Response Functions 
In view of the flexibility possible with active digital controllers, there is 
a very large number of possible over-all prototype response functions 
which can be implemented. As a starting point, however, the simplest, 

Fia. 7.3. Block diagram of sampled-data systems using digital controller. 
or “‘minimal,”’ prototype response functions’ are convenient. Minimal 
prototype systems are approached from the viewpoint that they must be 
able to respond satisfactorily to some convenient test input such as a step, 
ramp, or constant acceleration, or all three. The requirements which are 
set for minimal response functions are: 
1. The over-all response and the response of all elements of the system 
must be physically realizable. 
2. The steady-state response to the test input must have zero system- 
atic error. 
3. The transient response should be as fast as possible and the settling 
time should be equal to a finite number of sampling intervals. 
As an aid to applying these requirements, reference is made to the 
system in Fig. 7.3. Here, both D(z) and G(z) must be physically realiz- 
able, as must the over-all pulse transfer function K(z). Taking G(z) first, 
it can be assumed that it is a ratio of polynomials in z—! of the form 
Ge) ese Ta cea (7.6) 
The presence of go in this form is always assured if the system is physically 
realizable. This is shown readily by expanding G(z) into increasing 
